Dual of Constant Overlap-Add
In this section, we will derive the Fourier dual of the Constant OverLap-Add (COLA) condition for STFT analysis windows (discussed in §7.1). Recall that for perfect reconstruction using a hop-size of samples, the window must be . We will find that the equivalent frequency-domain condition is that the window transform must have spectral zeros at all frequencies which are a nonzero multiple of . Following established nomenclature for filter banks, we will say that such a window transform is .
Setting (the FFT hop size) gives
where (harmonics of the frame rate).
Looking across the top of Fig.8.16, for the case of input signal we have
Looking across the bottom of the figure, for the case of input signal
we have the output signal
This second form follows from the fact that complex sinusoids are eigenfunctions of linear systems--a basic result from linear systems theory [264,263].
Since the inputs were equal, the corresponding outputs must be equal too. This derives the Poisson Summation Formula (PSF):
Note that the PSF is the Fourier dual of the sampling theorem , [264, Appendix G].
The continuous-time PSF is derived in §B.15.
Thanks to the PSF, we may now express the COLA constraint in the frequency domain:
In other terms,
The ``Nyquist( )'' property for a function simply means that is zero at all nonzero multiples of (all harmonics of the frame rate here).
We may also refer to (8.33) as the ``weak COLA constraint'' in the frequency domain. It gives necessary and sufficient conditions for perfect reconstruction in overlap-add FFT processors. However, when the short-time spectrum is being modified, these conditions no longer apply, and a stronger COLA constraint is preferable.
An overly strong (but sufficient) condition is to require that the window transform be bandlimited consistent with downsampling by :
This condition is sufficient, but not necessary, for perfect OLA reconstruction. Strong COLA implies weak COLA, but it cannot be achieved exactly by finite-duration window functions.
When either of the strong or weak COLA conditions are met, we have
That is, the overlap-add of the window at hop-size is equal numerically to the dc gain of the window divided by .
Above, we used the Poisson Summation Formula to show that the constant-overlap-add of a window in the time domain is equivalent to the condition that the window transform have zero-crossings at all harmonics of the frame rate. In this section, we look briefly at the dual case: If the window transform is COLA in the frequency domain, what is the corresponding property of the window in the time domain? As one should expect, being COLA in the frequency domain corresponds to having specific uniform zero-crossings in the time domain.
Bandpass filters that sum to a constant provides an ideal basis for a graphic equalizer. In such a filter bank, when all the ``sliders'' of the equalizer are set to the same level, the filter bank reduces to no filtering at all, as desired.
Let denote the number of (complex) filters in our filter bank, with pass-bands uniformly distributed around the unit circle. (We will be using an FFT to implement such a filter bank.) Denote the frequency response of the ``dc channel'' by . Then the constant overlap-add property of the -channel filter bank can be expressed as
where as usual. By the dual of the Poisson summation formula, we have
where means that is zero at all nonzero integer multiples of , i.e.,
Thus, using the dual of the PSF, we have found that a good -channel equalizer filter bank can be made using bandpass filters which have zero-crossings at multiples of samples, because that property guarantees that the filter bank sums to a constant frequency response when all channel gains are equal.
The duality introduced in this section is the basis of the Filter-Bank Summation (FBS) interpretation of the short-time Fourier transform, and it is precisely the Fourier dual of the OverLap-Add (OLA) interpretation . The FBS interpretation of the STFT is the subject of Chapter 9.
Using ``square-root windows'' in the WOLA context, the valid hop sizes are identical to those for in the OLA case. More generally, given any window for use in a WOLA system, it is of interest to determine the hop sizes which yield perfect reconstruction.
Recall that, by the Poisson Summation Formula (PSF),
For WOLA, this is easily modified to become
where is the analysis window and is the synthesis window.
When , this becomes
In a weighted overlap-add system, the following windows can be used to satisfy the constant-overlap-add condition:
- For the rectangular window,
is a sinc function which reduces to
- For the Hamming window, the critically sampled window transform
has three nonzero samples (where the rectangular-window transform has
nonzero samples at critical
sampling. Measuring main-lobe width from zero-crossing to
zero-crossing as usual, we get
radians per sample, or
``6 side lobes'', for the width of
- The squared-Blackman window transform width is
- The square of a length
-term Blackman-Harris-family window
(where rect is
, Hann is
, etc.) has a main lobe of width
, measured from zero-crossing to zero-crossing in
``side-lobe units'' (
). This is up from
- The width of the main lobe can be used to determine the
hop size in the STFT, as will be discussed further in
Note that we need only find the first zero-crossing in the window transform for any member of the Blackman-Harris window family (Chapter 3), since nulls at all harmonics of that frequency will always be present (at multiples of ).
Convolving with Long Signals